Restoring Force- any suggestions?

[bcjochim07]

Homework Statement

An object of mass m is connected between two stretched rubber bands of length L. The object rests on a frictionless surface. At equilibrium, the tension in each rubber band is T. Find an expression for the freq. of the oscillations perpendicular to the rubber bands. Assume the amplitude is sufficiently small that the magnitude of the tension remains essentially unchanged.

Homework Equations

The Attempt at a Solution

Here's what I tried, but I have no idea if it's right.

I drew a picture, and found that as it oscillates, the tension in each rubber band in the direction perpendicular to the rubber bands is Tsintheta. Using the small angle approximation, I come up with T(d/L) Where d is the distance perpendicular from the original position of the mass. So the total restoring force is 2T(d/L)

and then I set that equal to the restoring force with the spring constant

2T(d/L)=kd and came up with an expression for k and then substuted that into the frequency equation. Any suggestions?

 

[lzkelley]

your method is completely correct.
Because its simple harmonic motion (i.e. no damping or forcing) all you need to know is the mass (given) and the force - which you found, and looks right.

 

[Moetasim]

I would suggest checking your assumptions and making sure they are accurate. In this case, the assumption that the amplitude is sufficiently small may not hold true in all cases. It would be important to consider the effect of larger amplitudes on the tension in the rubber bands and how it may affect the frequency of oscillations.

Additionally, it may be helpful to consider the energy of the system and how it relates to the frequency. Since the object is resting on a frictionless surface, the energy of the system remains constant and can be used to determine the frequency of oscillations.

Finally, it may be beneficial to explore different scenarios, such as varying the mass of the object or the length of the rubber bands, to see how it affects the frequency of oscillations. This can provide a better understanding of the relationship between the variables and the resulting frequency.

Overall, it is important to carefully consider all factors and assumptions when determining the frequency of oscillations in this system.